What is the solution of the differential equation `a (x"" (dy)/(dx) + 2y) = xy "" (dy)/(dx) `?

Options

A. `x^(2) = k y e^(y/a)`
B. `yx^(2) = k e^(y/a)`
C. `y^(2) x^(2) = k y e((y^2)/(a))`
D. None of these

Answer

To solve the differential equation \( a \left( x \frac{dy}{dx} + 2y \right) = xy \frac{dy}{dx} \), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ a \left( x \frac{dy}{dx} + 2y \right) = xy \frac{dy}{dx} \]
This can be rewritten as:
\[ a x \frac{dy}{dx} + 2a y = xy \frac{dy}{dx} \]

### Step 2: Collecting Terms
Next, we collect the terms involving \(\frac{dy}{dx}\):
\[ a x \frac{dy}{dx} - xy \frac{dy}{dx} = -2a y \]
Factoring out \(\frac{dy}{dx}\) gives us:
\[ \left( ax - xy \right) \frac{dy}{dx} = -2ay \]

### Step 3: Isolating \(\frac{dy}{dx}\)
Now, we isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{-2ay}{ax - xy} \]

### Step 4: Separating Variables
We can separate the variables \(y\) and \(x\):
\[ \frac{dy}{y} = \frac{-2a}{ax - xy} dx \]

### Step 5: Simplifying the Right Side
We can simplify the right side:
\[ \frac{dy}{y} = \frac{-2a}{x(a - y)} dx \]

### Step 6: Integrating Both Sides
Now we integrate both sides:
\[ \int \frac{dy}{y} = -2a \int \frac{1}{x(a - y)} dx \]
The left side integrates to:
\[ \ln |y| = -2a \ln |x| + C \]

### Step 7: Exponentiating Both Sides
Exponentiating both sides gives us:
\[ |y| = e^{C} |x|^{-2a} \]
Let \(k = e^{C}\), then:
\[ y = k |x|^{-2a} \]

### Step 8: Final Form
Thus, we can write the solution as:
\[ xy^2 = k e^{\frac{y}{a}} \]

### Conclusion
The solution to the differential equation is:
\[ x y^2 = k e^{\frac{y}{a}} \]

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